A103978 Expansion of (sqrt(1-12*x^2)+12*x^2+2*x-1)/(2*x*sqrt(1-12*x^2)).
1, 3, 6, 9, 54, 54, 540, 405, 5670, 3402, 61236, 30618, 673596, 288684, 7505784, 2814669, 84440070, 28146690, 956987460, 287096238, 10909657044, 2975361012, 124965162504, 31241290626, 1437099368796, 331638315876, 16581915793800
Offset: 0
Links
- Robert Israel, Table of n, a(n) for n = 0..1855
Programs
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Maple
rec:= -(n+1)*a(n)+2*(n-1)*a(n-1)+12*(2*n-3)*a(n-2)+24*(2-n)*a(n-3)+144*(4-n)*a(n-4): f:= gfun:-rectoproc({rec=0,a(0) = 1, a(1) = 3, a(2) = 6, a(3) = 9},a(n),remember): map(f, [$0..30]); # Robert Israel, Sep 13 2020 -
Mathematica
CoefficientList[Series[(Sqrt[1-12x^2]+12x^2+2x-1)/(2x Sqrt[1-12x^2]),{x,0,30}],x] (* Harvey P. Dale, Aug 06 2022 *)
Formula
G.f.: 1/sqrt(1-12*x^2)+(1-sqrt(1-12*x^2))/(2*x).
a(n) = sum{k=0..floor(n/2), 3^(n-k) * A000108(k) * C(k+1, n-k)}.
D-finite with recurrence: -(n+1)*a(n)+2*(n-1)*a(n-1) +12*(2n-3)*a(n-2) +24(2-n)*a(n-3) + 144*(4-n)*a(n-4)=0. - R. J. Mathar, Dec 14 2011
a(n) ~ 2^(n + 1/2) * 3^(n/2) / sqrt(Pi*n) if n is even and a(n) ~ 2^(n + 1/2) * 3^((n+1)/2) / (sqrt(Pi) * n^(3/2)) if n is odd. - Vaclav Kotesovec, Nov 19 2021